Gaokao Mathematics key terms

If every element of set A also belongs to set B, then A is a subset of B, written A⊆B. The empty set is a subset of every set, and every set is a subset of itself.

The set of all elements that belong to both set A and set B, written A∩B. It is empty exactly when the two sets share no common element, which is the usual test for disjointness.

Condition p is sufficient for conclusion q when p being true guarantees that q is true, that is p⇒q. Sufficiency concerns only the direction from hypothesis to conclusion, not the reverse.

p is a necessary and sufficient condition for q when both p⇒q and q⇒p hold, that is p⇔q. The two statements are then logically equivalent and may replace each other in reasoning.

The negation of 'for all x in M, p(x)' is 'there exists x in M such that not p(x)'. The quantifier changes type while the predicate is negated; the original domain M stays unchanged.

The set of all input values for which a function is defined and produces a real output. It must exclude values making denominators zero, even roots of negatives, or logarithms of non-positive numbers.

A function is increasing on an interval if larger inputs give larger outputs there, and decreasing if larger inputs give smaller outputs. Monotonicity on intervals is decided by the sign of the derivative.

The derivative f′(x₀) is the limit of the difference quotient as the increment tends to zero; geometrically it is the slope of the tangent line at that point and measures the instantaneous rate of change.

A local maximum or minimum of a function at an interior point where the derivative is zero or does not exist and the sign of the derivative changes. Extrema are function values, not the points themselves.

For a curve y=f(x) at the point (x₀,f(x₀)), the tangent line is y−f(x₀)=f′(x₀)(x−x₀). The point of tangency must lie on both the curve and the line.

For y=f(g(x)), the derivative is y′=f′(g(x))·g′(x). Differentiate the outer function evaluated at the inner function, then multiply by the derivative of the inner function.

In triangle ABC, a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius. It converts side problems into angle problems and back, and yields the circumradius directly.

In triangle ABC, a²=b²+c²−2bc·cos A, with the other two forms obtained by relabelling. It ties three sides to one angle and determines a missing side or angle in most configurations.

a sin x + b cos x can be written as √(a²+b²)·sin(x+φ), where the auxiliary angle φ satisfies tan φ = b/a and its quadrant is chosen to match the signs of a and b.

Formulas expressing trigonometric values of angles such as −x, π±x, π/2±x in terms of values at x. They follow the rule 'odd multiples of π/2 change the function name, even multiples keep it', with a sign from the quadrant.

The smallest positive T with f(x+T)=f(x) for all x in the domain. For sin and cos the period is 2π/|ω| and for tan it is π/|ω| when the argument is ωx plus a constant.

A sequence where each term after the first equals the previous term plus a fixed common difference d, so aₙ=a₁+(n−1)d. Its sum is n(a₁+aₙ)/2, and terms equally spaced from the ends have equal sums.

A sequence where each term after the first equals the previous term multiplied by a fixed nonzero common ratio q, so aₙ=a₁q^(n−1). The sum formula differs for q=1 and for q≠1, which must be discussed separately.

The sum Sₙ=a₁+a₂+…+aₙ of the first n terms of a sequence. It relates to the general term by aₙ=Sₙ−Sₙ₋₁ for n≥2, with a₁=S₁ checked separately.

A summation technique for sequences of the form (arithmetic)×(geometric). Multiply Sₙ by the common ratio, subtract the two expressions, and the middle terms collapse into a geometric series.

For positive reals a and b, (a+b)/2 ≥ √(ab), with equality exactly when a=b. It bounds sums below by products and products above by sums, and underlies many maximum–minimum problems.

A problem requiring an inequality to hold for every value of a variable in a given range. It is usually converted into a maximum or minimum condition on one side, or solved by separating the parameter.

For vectors a and b, a·b=|a||b|cos θ, which is a real number, not a vector. It gives the cosine of the included angle and vanishes exactly when the vectors are perpendicular and nonzero.

For z=a+bi, the modulus |z|=√(a²+b²) is the distance from the origin to the point (a,b) in the complex plane. It satisfies |z₁z₂|=|z₁||z₂| and |z|²=z·z̄, a real number.

For z=a+bi, the conjugate z̄=a−bi, obtained by reversing the sign of the imaginary part. Their product is the real number a²+b², which is used to rationalise complex denominators.

A nonzero vector perpendicular to a plane. Together with one point it determines the plane, and it turns angle and distance questions involving planes into dot-product computations.

The angle between two half-planes sharing an edge, measured by the angle between two rays perpendicular to the edge, one in each face. The angle between the planes' normal vectors equals it or its supplement.

The angle between a line and a plane, defined as the complement of the angle between the line and a normal to the plane. Its sine equals |cos⟨direction vector, normal⟩|, and it lies in [0, π/2].

For an ellipse, e=c/a with 0<e<1, where c is the focal distance and a the semi-major axis. It measures flatness: values near 0 give nearly circular ellipses, values near 1 give very elongated ones.

The two lines y=±(b/a)x that the branches of x²/a²−y²/b²=1 approach but never meet. Their slopes control how widely the branches open and are fixed by the ratio b/a.

For y²=2px, the line x=−p/2. Every point of the parabola is equidistant from the focus (p/2,0) and this line, which is the definition used in most problems.

For a line cutting a conic at two points, |AB|=√(1+k²)·|x₁−x₂| where k is the slope. It reduces chord length to a difference of roots obtained from Vieta's formulas.

P(B|A)=P(AB)/P(A), the probability that B occurs given that A has occurred, defined only when P(A)>0. It restricts the sample space to A and requires probabilities to be recomputed inside that reduced space.

For a discrete random variable, E(X)=Σxᵢpᵢ, the probability-weighted average of its values. It is linear, so E(aX+b)=aE(X)+b, it need not be a possible value of X, and it describes the long-run average outcome.

The distribution of the number of successes in n independent trials, each with the same success probability p, written X~B(n,p). Its probabilities are P(X=k)=C(n,k)p^k(1−p)^(n−k) and its mean is np.

An ordering of k distinct objects chosen from n different ones, counted by A(n,k)=n!/(n−k)!. Order matters, so the same objects arranged differently count as different outcomes.

A selection of k objects from n different ones with order ignored, counted by C(n,k)=n!/[k!(n−k)!]. Each combination corresponds to k! permutations of the same objects, which links the two counting formulas.

(a+b)ⁿ=Σ C(n,k)a^(n−k)b^k for k=0,1,…,n. It expands a binomial power without multiplying everything out, and its general term locates any required term or coefficient.